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Matching statistic: St000142
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Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[1,-2] => [1,-2] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[2,-1] => [-1,2] => [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1
[1,-3,-2] => [1,-2,-3] => [1]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> 1
[2,-1,3] => [-1,2,-3] => [1]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> 1
[3,-1,2] => [-1,-2,3] => [1]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 0
Description
The number of even parts of a partition.
Matching statistic: St000155
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00163: Signed permutations —permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[1,-2] => [1,-2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[2,-1] => [-1,2] => [1,2] => 0
[-2,1] => [-2,-1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,-3] => [1,2,3] => 0
[1,-2,3] => [1,-2,-3] => [1,2,3] => 0
[1,-2,-3] => [1,-2,-3] => [1,2,3] => 0
[-1,2,3] => [-1,-2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[1,3,-2] => [1,-2,3] => [1,2,3] => 0
[1,-3,2] => [1,-3,-2] => [1,3,2] => 1
[1,-3,-2] => [1,-2,-3] => [1,2,3] => 0
[-1,3,2] => [-1,-2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,-3] => [2,1,-3] => [2,1,3] => 1
[2,-1,3] => [-1,2,-3] => [1,2,3] => 0
[2,-1,-3] => [-1,2,-3] => [1,2,3] => 0
[-2,1,3] => [-2,-1,3] => [2,1,3] => 1
[-2,1,-3] => [-2,-1,-3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,-1] => [-1,2,3] => [1,2,3] => 0
[2,-3,1] => [-3,2,-1] => [3,2,1] => 1
[2,-3,-1] => [-1,2,-3] => [1,2,3] => 0
[-2,3,1] => [-2,-1,3] => [2,1,3] => 1
[-2,-3,1] => [-2,-1,-3] => [2,1,3] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,1,-2] => [3,-2,1] => [3,2,1] => 1
[3,-1,2] => [-1,-2,3] => [1,2,3] => 0
[3,-1,-2] => [-1,-2,3] => [1,2,3] => 0
[-3,1,2] => [-3,2,-1] => [3,2,1] => 1
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,-1] => [-1,3,2] => [1,3,2] => 1
[3,-2,1] => [-2,-1,3] => [2,1,3] => 1
[3,-2,-1] => [-1,-2,3] => [1,2,3] => 0
[-3,2,1] => [-3,2,-1] => [3,2,1] => 1
[-3,2,-1] => [-1,-3,-2] => [1,3,2] => 1
[-3,-2,1] => [-2,-1,-3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,2,3,4] => 0
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000162
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00163: Signed permutations —permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[1,-2] => [1,-2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[2,-1] => [-1,2] => [1,2] => 0
[-2,1] => [-2,-1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,-3] => [1,2,3] => 0
[1,-2,3] => [1,-2,-3] => [1,2,3] => 0
[1,-2,-3] => [1,-2,-3] => [1,2,3] => 0
[-1,2,3] => [-1,-2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[1,3,-2] => [1,-2,3] => [1,2,3] => 0
[1,-3,2] => [1,-3,-2] => [1,3,2] => 1
[1,-3,-2] => [1,-2,-3] => [1,2,3] => 0
[-1,3,2] => [-1,-2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,-3] => [2,1,-3] => [2,1,3] => 1
[2,-1,3] => [-1,2,-3] => [1,2,3] => 0
[2,-1,-3] => [-1,2,-3] => [1,2,3] => 0
[-2,1,3] => [-2,-1,3] => [2,1,3] => 1
[-2,1,-3] => [-2,-1,-3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,-1] => [-1,2,3] => [1,2,3] => 0
[2,-3,1] => [-3,2,-1] => [3,2,1] => 1
[2,-3,-1] => [-1,2,-3] => [1,2,3] => 0
[-2,3,1] => [-2,-1,3] => [2,1,3] => 1
[-2,-3,1] => [-2,-1,-3] => [2,1,3] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,1,-2] => [3,-2,1] => [3,2,1] => 1
[3,-1,2] => [-1,-2,3] => [1,2,3] => 0
[3,-1,-2] => [-1,-2,3] => [1,2,3] => 0
[-3,1,2] => [-3,2,-1] => [3,2,1] => 1
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,-1] => [-1,3,2] => [1,3,2] => 1
[3,-2,1] => [-2,-1,3] => [2,1,3] => 1
[3,-2,-1] => [-1,-2,3] => [1,2,3] => 0
[-3,2,1] => [-3,2,-1] => [3,2,1] => 1
[-3,2,-1] => [-1,-3,-2] => [1,3,2] => 1
[-3,-2,1] => [-2,-1,-3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,2,3,4] => 0
Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000703
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00163: Signed permutations —permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[1,-2] => [1,-2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[2,-1] => [-1,2] => [1,2] => 0
[-2,1] => [-2,-1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,-3] => [1,2,3] => 0
[1,-2,3] => [1,-2,-3] => [1,2,3] => 0
[1,-2,-3] => [1,-2,-3] => [1,2,3] => 0
[-1,2,3] => [-1,-2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[1,3,-2] => [1,-2,3] => [1,2,3] => 0
[1,-3,2] => [1,-3,-2] => [1,3,2] => 1
[1,-3,-2] => [1,-2,-3] => [1,2,3] => 0
[-1,3,2] => [-1,-2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,-3] => [2,1,-3] => [2,1,3] => 1
[2,-1,3] => [-1,2,-3] => [1,2,3] => 0
[2,-1,-3] => [-1,2,-3] => [1,2,3] => 0
[-2,1,3] => [-2,-1,3] => [2,1,3] => 1
[-2,1,-3] => [-2,-1,-3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,-1] => [-1,2,3] => [1,2,3] => 0
[2,-3,1] => [-3,2,-1] => [3,2,1] => 1
[2,-3,-1] => [-1,2,-3] => [1,2,3] => 0
[-2,3,1] => [-2,-1,3] => [2,1,3] => 1
[-2,-3,1] => [-2,-1,-3] => [2,1,3] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,1,-2] => [3,-2,1] => [3,2,1] => 1
[3,-1,2] => [-1,-2,3] => [1,2,3] => 0
[3,-1,-2] => [-1,-2,3] => [1,2,3] => 0
[-3,1,2] => [-3,2,-1] => [3,2,1] => 1
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,-1] => [-1,3,2] => [1,3,2] => 1
[3,-2,1] => [-2,-1,3] => [2,1,3] => 1
[3,-2,-1] => [-1,-2,3] => [1,2,3] => 0
[-3,2,1] => [-3,2,-1] => [3,2,1] => 1
[-3,2,-1] => [-1,-3,-2] => [1,3,2] => 1
[-3,-2,1] => [-2,-1,-3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,2,3,4] => 0
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000994
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00163: Signed permutations —permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[1,-2] => [1,-2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[2,-1] => [-1,2] => [1,2] => 0
[-2,1] => [-2,-1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,-3] => [1,2,3] => 0
[1,-2,3] => [1,-2,-3] => [1,2,3] => 0
[1,-2,-3] => [1,-2,-3] => [1,2,3] => 0
[-1,2,3] => [-1,-2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[1,3,-2] => [1,-2,3] => [1,2,3] => 0
[1,-3,2] => [1,-3,-2] => [1,3,2] => 1
[1,-3,-2] => [1,-2,-3] => [1,2,3] => 0
[-1,3,2] => [-1,-2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,-3] => [2,1,-3] => [2,1,3] => 1
[2,-1,3] => [-1,2,-3] => [1,2,3] => 0
[2,-1,-3] => [-1,2,-3] => [1,2,3] => 0
[-2,1,3] => [-2,-1,3] => [2,1,3] => 1
[-2,1,-3] => [-2,-1,-3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,-1] => [-1,2,3] => [1,2,3] => 0
[2,-3,1] => [-3,2,-1] => [3,2,1] => 1
[2,-3,-1] => [-1,2,-3] => [1,2,3] => 0
[-2,3,1] => [-2,-1,3] => [2,1,3] => 1
[-2,-3,1] => [-2,-1,-3] => [2,1,3] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,1,-2] => [3,-2,1] => [3,2,1] => 1
[3,-1,2] => [-1,-2,3] => [1,2,3] => 0
[3,-1,-2] => [-1,-2,3] => [1,2,3] => 0
[-3,1,2] => [-3,2,-1] => [3,2,1] => 1
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,-1] => [-1,3,2] => [1,3,2] => 1
[3,-2,1] => [-2,-1,3] => [2,1,3] => 1
[3,-2,-1] => [-1,-2,3] => [1,2,3] => 0
[-3,2,1] => [-3,2,-1] => [3,2,1] => 1
[-3,2,-1] => [-1,-3,-2] => [1,3,2] => 1
[-3,-2,1] => [-2,-1,-3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,2,3,4] => 0
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St001251
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[1,-2] => [1,-2] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[2,-1] => [-1,2] => [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1
[1,-3,-2] => [1,-2,-3] => [1]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> 1
[2,-1,3] => [-1,2,-3] => [1]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> 1
[3,-1,2] => [-1,-2,3] => [1]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 0
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[1,-2] => [1,-2] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[2,-1] => [-1,2] => [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1
[1,-3,-2] => [1,-2,-3] => [1]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> 1
[2,-1,3] => [-1,2,-3] => [1]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> 1
[3,-1,2] => [-1,-2,3] => [1]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 0
Description
Half the sum of the even parts of a partition.
Matching statistic: St001269
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00163: Signed permutations —permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[1,-2] => [1,-2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[2,-1] => [-1,2] => [1,2] => 0
[-2,1] => [-2,-1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,-3] => [1,2,3] => 0
[1,-2,3] => [1,-2,-3] => [1,2,3] => 0
[1,-2,-3] => [1,-2,-3] => [1,2,3] => 0
[-1,2,3] => [-1,-2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[1,3,-2] => [1,-2,3] => [1,2,3] => 0
[1,-3,2] => [1,-3,-2] => [1,3,2] => 1
[1,-3,-2] => [1,-2,-3] => [1,2,3] => 0
[-1,3,2] => [-1,-2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,-3] => [2,1,-3] => [2,1,3] => 1
[2,-1,3] => [-1,2,-3] => [1,2,3] => 0
[2,-1,-3] => [-1,2,-3] => [1,2,3] => 0
[-2,1,3] => [-2,-1,3] => [2,1,3] => 1
[-2,1,-3] => [-2,-1,-3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,-1] => [-1,2,3] => [1,2,3] => 0
[2,-3,1] => [-3,2,-1] => [3,2,1] => 1
[2,-3,-1] => [-1,2,-3] => [1,2,3] => 0
[-2,3,1] => [-2,-1,3] => [2,1,3] => 1
[-2,-3,1] => [-2,-1,-3] => [2,1,3] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 1
[3,1,-2] => [3,-2,1] => [3,2,1] => 1
[3,-1,2] => [-1,-2,3] => [1,2,3] => 0
[3,-1,-2] => [-1,-2,3] => [1,2,3] => 0
[-3,1,2] => [-3,2,-1] => [3,2,1] => 1
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 1
[3,2,-1] => [-1,3,2] => [1,3,2] => 1
[3,-2,1] => [-2,-1,3] => [2,1,3] => 1
[3,-2,-1] => [-1,-2,3] => [1,2,3] => 0
[-3,2,1] => [-3,2,-1] => [3,2,1] => 1
[-3,2,-1] => [-1,-3,-2] => [1,3,2] => 1
[-3,-2,1] => [-2,-1,-3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,4] => 0
[1,2,-3,4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,2,3,4] => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,2,3,4] => 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,2,3,4] => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,2,3,4] => 0
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[1,-2] => [1,-2] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[2,-1] => [-1,2] => [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1
[1,-3,-2] => [1,-2,-3] => [1]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> 1
[2,-1,3] => [-1,2,-3] => [1]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> 1
[3,-1,2] => [-1,-2,3] => [1]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001657
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[1,-2] => [1,-2] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[2,-1] => [-1,2] => [1]
=> 0
[-2,1] => [-2,-1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,-3] => [1,2,-3] => [1,1]
=> 0
[1,-2,3] => [1,-2,-3] => [1]
=> 0
[1,-2,-3] => [1,-2,-3] => [1]
=> 0
[-1,2,3] => [-1,-2,3] => [1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[1,3,-2] => [1,-2,3] => [1,1]
=> 0
[1,-3,2] => [1,-3,-2] => [2,1]
=> 1
[1,-3,-2] => [1,-2,-3] => [1]
=> 0
[-1,3,2] => [-1,-2,3] => [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,1,-3] => [2,1,-3] => [2]
=> 1
[2,-1,3] => [-1,2,-3] => [1]
=> 0
[2,-1,-3] => [-1,2,-3] => [1]
=> 0
[-2,1,3] => [-2,-1,3] => [2,1]
=> 1
[-2,1,-3] => [-2,-1,-3] => [2]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> 1
[2,3,-1] => [-1,2,3] => [1,1]
=> 0
[2,-3,1] => [-3,2,-1] => [2,1]
=> 1
[2,-3,-1] => [-1,2,-3] => [1]
=> 0
[-2,3,1] => [-2,-1,3] => [2,1]
=> 1
[-2,-3,1] => [-2,-1,-3] => [2]
=> 1
[3,1,2] => [3,2,1] => [2,1]
=> 1
[3,1,-2] => [3,-2,1] => [2]
=> 1
[3,-1,2] => [-1,-2,3] => [1]
=> 0
[3,-1,-2] => [-1,-2,3] => [1]
=> 0
[-3,1,2] => [-3,2,-1] => [2,1]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[3,2,-1] => [-1,3,2] => [2]
=> 1
[3,-2,1] => [-2,-1,3] => [2,1]
=> 1
[3,-2,-1] => [-1,-2,3] => [1]
=> 0
[-3,2,1] => [-3,2,-1] => [2,1]
=> 1
[-3,2,-1] => [-1,-3,-2] => [2]
=> 1
[-3,-2,1] => [-2,-1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 0
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
=> 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
=> 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
=> 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
=> 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
=> 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
=> 0
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000185The weighted size of a partition. St000211The rank of the set partition. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000884The number of isolated descents of a permutation. St001176The size of a partition minus its first part. St001489The maximum of the number of descents and the number of inverse descents. St001737The number of descents of type 2 in a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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